What's the term for death by dissolving in AI?
I find the concept of the a Turing machine useful. In one dimension, everything is a string. All of the parts that are "not you" are merely a substrate, a medium for the program your_mind runs on top of. The you, your identity, the "metaphysical" component we think of as the mind, is a result of running the algorithm that is your_mind on the bioware of your body, or, the hardware (technically, "wetware".) So what we're really talking about is the software, and in that light I might use: 


Translation


because the software is being translated to execute in a new environment, or


Migration


as in moving software from one system to another.

Philip Dick wrote a philosophical narrative, not technically sci-fi, called The Transmigration of Timothy Archer which is about identity moving between bodies. In a rare departure from his usual work about AI and the effects of a technological society on the human spirit, this book looks at the question of identity in the context of the soul, which opens up all kinds of philosophical questions surrounding the type of technology we're speculating on, particularly in relation to the self.   

I value artistic insight, and Phillip K. is considered quite prescient, so perhaps


Transmigration


is most appropriate, as it carries both metaphysical and information technology meanings.



Re death by dissolution, it's worthwhile to look at the etymology of dissolve: 


  late 14c. (transitive and intransitive) "to break up" (of material substances), from Latin dissolvere "to loosen up, break apart," from dis- "apart" (see dis-) + solvere "to loosen, untie," from PIE *se-lu-, from reflexive pronoun *s(w)e- (see idiom) + root *leu- "to loosen, divide, cut apart." Meaning "to disband" (an assembly) is early 15c. Related: Dissolved; dissolving.


I think you're on the right track with dilution, certainly per the modern usage, but it may be possible to get more precise.

It's not about breaking up, or washing away (except metaphorically); rather, it's about minimization as in the diminishment of the original software kernel (the self) in relation to an expanding algorithm.


  early 15c., from merger of two obsolete verbs, diminue and minish. Diminue is from Old French diminuer "make small," from Latin diminuere "break into small pieces," variant of deminuere "lessen, diminish," from de- "completely" + minuere "make small" (from PIE root *mei- (2) "small").


The Old French diminuer is apt, as is the Latin deminuere and deminuo, which also carries a meaning of "civil death" and "abatement". Abatement carries a meaning of mitigation, which can be defined fundamentally as "lessening of effect," in this case of the kernel of the original self in relation to the new aggregate.

It may be useful to think of it as a ratio 1/ℵ, with the self as the 1. Important to note the ratio is not literal--each number represent an aggregation of functions we call programs. 

That's how I'd think of it mathematically, but metaphysically, Nat nails it by referencing Jung and the death of the ego.

Ego comes from the Latin noun for the self (I, me) which can also be plural (we, us).  It's also fun to note that the Latin verb of being is sum, because in English, sum has a mathematical meaning of an aggregate considered as a whole. 

Because we're talking about what it means to be a person, but we also want to connote a function (a process of relative minimization) I am thinking:


Depersonalization



  (noun) the action of divesting someone or something of human characteristics or individuality.SOURCE: Google


The fun part is that there's a legal definition of person:


  In legal use, "corporate body or corporation having legal rights," 15c., short for person aggregate (c. 1400), person corporate (mid-15c.)


Which I think applies (loosely) to the aggregation of function that comprises the program we call the self, and the greater aggregation of functions that form a new self.  

Because here we're talking about the loss of the original individual, depersonalization as opposed to repersonalization as a corporate entity. 

Alternately, I might propose:


Deindividualization 


with the definition "a loss in individual identity within a group" (also known as deindividuation).

(Charles Stross wrote about this in Accellerando, where a character chooses to be subsumed by a group intelligence as the only means of escaping a swarm of autonomous lawsuits;)